Is it consistent that there exists a nonzero atomless finite measure on some $\sigma$-algebra on a cardinal $\kappa$ satisfying $\kappa<\mathfrak{c}$? Can there be such a measure on $\omega_1$ when $\omega_1<\mathfrak{c}$?

I would generally like to know where one can find related results.

My motivation comes from people in mathematical economics using the term "continuum of agents" for models in which agents are modeled by nonatomic probability spaces. I'd like to know if there couldn't be fewer agents than a continuum with this formulation.